Investigation of Compressible Viscous Flows over Open Cavities Including the Effects of Shear Layer Thickness and Multiple Deck Structure on Interaction with the Trailing Edge

The problem considered in this paper is that of compressible viscous flow over an open rectangular cavity, including the effects of shear layer thickness and triple deck structure on interaction with the trailing edge. This analysis unifies analytical cavity studies and provides a wave-number correction for the method of Tam and Block [1], which studies the case of a compressible inviscid flow with a constant shear layer thickness spanning an open cavity. Here, basic equations for a two-dimensional compressible viscous flow are derived. The effect of non-parallelism of the mean flow is introduced. This weakly non-parallel mean flow is perturbed to obtain the governing equations for the shear layer. The inverse of the Reynolds number, here related to the weak effect of non-parallelism of the mean flow, serves as a perturbation parameter. The shear layer is divided into a region of inviscid flow away from the trailing edge and a region where viscous effects are important, near the trailing edge, as a typical boundary layer problem. The viscous region is analyzed by proper scaling of the independent variables. Distinguished limits are obtained by balancing the terms in the set of the governing equations. A multiple deck structure, containing three distinct regions, occurs, each region with a different scaling. Normal modes for the flow properties are introduced to predict an eigenvalue problem, which governs the wave number/frequency relationship in each deck, as well as the inviscid region. These eigenvalue problems are derived using the Fredholm Alternative and are solved numerically by use of a fourth-order Runge-Kutta method. The method of asymptotic expansions is used to match these wave numbers to those among the multiple deck structure, as well as to the one for the inviscid region. This wave number, uniformly valid throughout the region, is used as a correction to the one derived by Tam and Block. This study proves that by considering the effect of non-parallelism of the mean flow, a lower wave-number/frequency is produced at any spanwise location for a given excitation frequency. Predicted discrete tone frequencies, based on this corrected value of wave-number, produce a closer agreement with the experimental results.